Instability Thresholds and Dynamics of Mesa Patterns in Reaction-Diffusion Systems

Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems Rebecca McKay and Theodore Kolokolnikov Department of Mathematics and Statistics, Dalhousie University, Canada Abstract We consider a class of one-dimensional reaction-diffusion systems, u = ε2u + f(u, w) t xx . 0= Dw + g(u, w) xx Under some generic conditions on the nonlinearities f,g, in the singular limit ε 0, and for a fixed D independent of ε, such a system exhibits a steady state consisting→ of sharp back-to-back interfaces which is stable in time. On the other hand, it is also known that in the so-called shadow limit D = , the periodic pattern having more than one interface is unstable. In this paper, we analyse∞ in detail the transition between the stable patterns when D = O(1) and the shadow system when D = . We show that this transition occurs when D is exponentially large in ε and we derive instability∞ thresholds D D D . such 1 2 3 that a periodic pattern with 2K interfaces is stable if D<DK and is unstable when D>DK . We also study the dynamics of the interfaces when D is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed; excellent agreement with the asymptotic results is observed. 1 Introduction One of the most prevalent phenomena observed in reaction-diffusion systems is the formation of mesa patterns. Such patterns consist of a sequence of highly localized interfaces (or kinks) that are separated in space by regions where the solution is nearly constant. These patterns has been studied intensively for the last three decades and by now an extensive literature exists on this topic. We refer for example to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein. In this paper we are concerned with the following class of reaction-diffusion models: u = ε2u + f(u, w) t xx (1) 0= Dw + g(u, w) xx in the limit ε 1 and D 1. (2) Under certain general conditions on the nonlinearities f and g that will be specified below, the system (1) admits an interface layer solution. Such a solution has the property that u is either u+ or u− for some constants u = u− everywhere except near the interface location, where it has a layer of size O(ε) + 6 connecting the two constant states u±. A single mesa (or a box) solution consists of two interfaces, one connecting u− to u+ and another connecting u+ back to u−. By mirror reflection, a single mesa can be extended to a symmetric K mesa solution, consisting of K mesas or 2K interfaces (see Figure 1b). The general system (1) has been thoroughly studied by many authors. In particular, the regime D = O(1) is well understood. Under certain general conditions on g and f, it is known that a K-mesa 1 pattern is stable for all K – see for example [1], [11]. On the other hand, in the limit D = , the system (1) reduces to the so-called shadow system, ∞ 2 ut = ε uxx + f(u, w0); g(u, w0)=0. (3) Z Equation (3) also includes many models of phase separation such as Allen-Cahn [4] as a special case. Under the same general conditions on g and f, a single interface of the shadow system is also stable; however a pattern consisting of more than one interface is known to be unstable [12]. The main question that we address in this paper is how the transition from the stable regime D = O(1) to the unstable shadow regime D = takes place. In a nutshell, when D is not too large, the stability of K mesa pattern as shown in [1]∞ is due to the stabilizing effect of the global variable w. However as D is increased, the stabilizing effect of w is decreased, and eventually the pattern loses its stability. We find that the onset of instability is caused by the interaction between the interfaces of u. This interaction is exponentially small, but becomes important as D becomes exponentially large. By analysing the contributions to stability of both w and u, we compute an explicit sequence of threshold values D1 >D2 >D3 >... such that a K mesa solution on the domain of a fixed size 2R is stable if D<DK 1 and is unstable if D>DK. These thresholds have the order ln(DK )= O( ε ), so that DK is exponentially large in ε. 7 6 5 4 3 2 1 0 −1 0 1 2 3 4 5 6 7 (a) (b) Figure 1: (a) Coarsening process in Lengyel-Epstein model (8), starting with random initial conditions. Time evolution of u is shown; note the logarithmic time scale. Initial inhomogenuities are amplified due to Turing instability. Shortly after, localized structures are formed at around t = 10. This is followed by a long transient pattern consisting of three mesas. The middle mesa eventually disappears around t 100 due to the very slow instability of the three-mesa pattern. The remaining two mesas then move towards∼ symmetrical configuration which appears to be stable. The parameter values are ε = 0.06, a = 10, D = 500, τ = 0.1, domain size is 8. (b) The snapshot of u at t = 5, 000, 000, by which time the system has reached a steady state consisting of a two-mesa pattern. There are many systems that fall in the class described in this paper. Let us now mention some of them. Cubic model: One of the simplest systems is a cubic model, • u = ε2u +2u 2u3 + w t xx . (4) 0= Dw −u + β xx − 0 2 It is a variation on FitzHugh-Nagumo model used in [5] and in [13]. It is a convenient model for testing our asymptotic results. Model of Belousov-Zhabotinskii reaction in water-in-oil microemulsion: This model is a • was introduced in [14], [15]. In [16], the following simplified model was analysed in two dimensions: u q u = ε2u f − + wu u2 t xx − 0 u + q − . (5) 0= Dw +1 uw xx − Brusselator model. In [17] the authors considered coarsening phenomenon in the Brusselator • model; a self-replication phenomenon was studied in [11]. After a change of variables the Brusselator may written as u = ε2u u + uw u3 t xx . (6) 0= Dw − β u +1− xx − 0 Gierer-Meinhardt model with saturation. This model was introduced in [18], (see also [19], • [20]) to model stripe patterns on animal skins. After some rescaling, it is u2 u = ε2u u + t xx − w(1 + κu2) . (7) 0= Dw w + u2 xx − This model was also studied in [21], where stripe instability thresholds were computed. Lengyel-Epstein model. This model was introduced in [22], see also [23]: • 4uv u = u + a u t xx 1+ u2 − − uv . (8) τv = Dv + u t xx − 1+ u2 Other models include: models for co-existence of competing species [24]; vegetation patterns in • dry regions [25]; models of chemotaxis [26] and models of phase separation in diblock copolymers [27]. Figure 1(a) illustrates the instability phenomenon studied in this paper, as observed numerically in the Lengyel-Epstein model. Starting with random initial conditions, Turing instability leads to a formation of a three-mesa pattern at t 10. However such pattern is unstable, even though this only becomes apparent much later (at t ∼100). The resulting two-mesa pattern then drifts towards a symmetric position where it eventually∼ settles. A similar phenomenon for the Belousov-Zhabotinskii model (5) is illustrated in Figure 7(b). It shows the time-evolution a 2-mesa solution to (5) with D>D2, starting with initial conditions that consist of a slightly perturbed two-mesa pattern. After a very long time, one of the mesas absorbs the mass of the other, then moves towards the center of the domain where it remains as a stable pattern. In addition to studying the stability of the mesa patterns, we also study their dynamics. This allows us to describe in detail the mechanism by which the exchange of mass between two mesas can take place, as well as the motion of the interfaces away from the equilibrium. For a pattern consisting of K mesas, we derive a reduced problem, consisting of 2K ODE’s that gouvern the asymptotic motion of the 2K interfaces. The outline of the paper is as follows. In 2 we construct the steady state consisting of K mesas. The construction is summarized in Proposition§ 1. The main result is presented in 3 (Theorem 2), where we analyse the asyptotics of the linearized problem for the periodic pattern, and§ derive the instability thresholds DK . In 4 we derive the reduced equations of motion for the interfaces. In 5 we present numerical computations§ to support our asymptotic results. We conclude with discussion of open§ problems in 6. § 3 2 Preliminaries: construction of the K mesa steady state − We start by constructing the time-independent mesa-type solution to (1). The mesa (or box) solution consists of two back-to-back interfaces. Thus we first consider the conditions for existence of a single interface solution and review its construction. A mesa solution can then be constructed from a single interface by reflecting and doubling the domain size. Similarly, a K-mesa pattern is then constructed by making K copies of a single mesa. We summarize the construction as follows. Proposition 1 Consider the time-independent steady state of the PDE system (1) satisfying 0= ε2u + f(u, w) xx (9) 0= Dw + g(u, w) xx with Neumann boundary conditions and in the limit ε 1 and D 1.

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